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An analytical technique is provided to estimate queue-length and delay distributions for multi-queue systems using generalised processor sharing discipline with time-correlated variable service rates, based on two-dimensional multi-level decoupling. First, temporal decomposition is used to convert the time-correlated queuing problem into a set of sub-problems over several timescales. Subsequently, queue decomposition exploits the queue weight dependencies to convert a multi-queue problem into a set of single-queue problems. The core of the analysis lies in estimating the multi-scale service rate models for each of these queues. The authors show the hierarchy of this estimation and the dependency of the queue service rate on the unused capacity of the other queues and their weights. Simulation and analytical results on queue and delay survivor functions are in good agreement.